Travaux Pratiques 2: Risk Management
1 Loading the data
The file Market.rda contains the list Market.
The first element of the list Market is an xts object corresponding to the value of the SP500 index from “2000-01-03” until “2013-09-10”.
The second element vix corresponds to the value of the VIX index at the same dates.
The third element rf corresponds to the term structure on “2013-09-10” for 14 maturities varying from 1 day to 30 years. The term structure for the interest rates can be seen in Figure 1.1 .
Figure 1.1: Interest rates term structure.
The fourth (calls) and the fifth (puts) contains the strikes (K), maturities (tau) and implied volatilities (IV) for calls and puts options respectively.
We plot the volatility surface in Figure 1.2 .
Figure 1.2: Volatility surface for the options in the list Market.
2 Pricing of a portfolio of options
The prices of the call options using the Black-Scholes formula are given in table 2.1 .
| S0 | K | tau | sigma | r | C |
|---|---|---|---|---|---|
| 1683.99 | 1600 | 0.08 | 0.1453 | 0.00115 | 87.569 |
| 1683.99 | 1650 | 0.08 | 0.1453 | 0.00115 | 47.724 |
| 1683.99 | 1750 | 0.16 | 0.1453 | 0.00144 | 15.302 |
| 1683.99 | 1800 | 0.16 | 0.1453 | 0.00144 | 6.380 |
The price of this book (the sum of prices of the calls that constitutes the book) is \(V_0\) = 156.98.
3 One risk driver and Gaussian model
In this section, we assume that the risk of our book of options is driven by a single factor namely the change in price of the SP500 index. The log-returns of the index are assumed to follow a normal distribution with parameters given by the mean and standard deviation from the historical values.
Figure 3.1: P&L distribution for normal log-returns of the SP500. The red line at -122.77 is the value at risk at 95% probability.
We model 10000 possible values for the stocks in 5 days and use them to estimate the distribution for the P&L in a week (5 days). The distribution of the P&L is given in Figure 3.1 .
The VaR at risk at the 95% confidence level is VaR95 = -122.77 and the expected shortfall is ES95 = -134.95. So we expect the book of options to lose at least 122.77 from its initial value 5% of the time.
4 Two risk drivers and Gaussian model
In this section, we assume that the risk of our book of options is driven by two factors namely the change in price of the SP500 index and the volatility of the market (given by the VIX). The log-returns of the SP500 and the VIX are assumed to follow a normal distribution with parameters given by the mean and standard deviation from the historical values.
Figure 4.1: P&L distribution when log-returns of SP500 and VIX are assumed to follow a bivariate normal distribution. The red line at -111.59 represents the value at risk at 95% probability.
We model 10000 possible values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L in a week (5 days). The distribution of the P&L is given in Figure 4.1 .
The VaR at risk at the 95% confidence level is VaR95 = -111.59 and the expected shortfall is ES95 = -123.7. So we expect the book of options to lose at least 111.59 from its initial value 5% of the time.
5 Two risk drivers and copula-marginal model (Student-t and Gaussian copula)
In this section, we assume that the risk of our book of options is driven by two factors namely the change in price of the SP500 index and the volatility of the market (given by the VIX). The log-returns of the SP500 are assumed to follow a Student-t distribution with \(\nu = 10\) degrees of freedom and the log-returns of the VIX are assumed to follow a Student-t distribution with \(\nu = 5\) degrees of freedom. A normal Copula is assumed to merge the marginals.
Figure 5.1: P&L distribution when log-returns of the SP500 and the VIX are assumed to have Student-t marginals and their marginals to be linked by a Gaussian copula . The red line at -99.02 represents the value at risk at 95% probability.
We model 10000 possible values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L in a week (5 days). The distribution of the P&L is given in Figure 5.1 .
The VaR at risk at the 95% confidence level is VaR95 = -99.02 and the expected shortfall is ES95 = -113.06. So we expect the book of options to lose at least 99.02 from its initial value 5% of the time.
6 Volatility surface
Figure 6.1: P&L distribution when log-returns of the SP500 and the VIX are assumed to follow a bivariate normal distribution. The simulated values of the VIX are shifted by the one year ATM volatility difference. The red line at -108.6 represents the value at risk at 95% probability.
We model 10000 possible values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L in a week (5 days). The simulated values of the VIX are then shifted by the difference between the one year ATM value of the implied volatility and today value of the VIX. The distribution of the P&L is given in Figure 6.1 .
The VaR at risk at the 95% confidence level is VaR95 = -108.6 and the expected shortfall is ES95 = -121.15. So we expect the book of options to lose at least 108.6 from its initial value 5% of the time.
7 Full approach
Figure 7.1: P&L distribution when log-returns of the SP500 follow an ARMA(1,1) and the log-returns of the VIX an AR(1). The invariants of the two processes are linked by a Gaussian copula. The red line at -71.41 represents the value at risk at 95% probability.
Here, we fit a GARCH(1,1) to the log-returns of the SP500 and an AR(1) process to the log-returns of the VIX. The residuals of the two processes are assumed to be invariants and to be linked by a Gaussian copula. We generate 10000 possible values for the SP500 and the VIX in 5 days and use them to estimate the distribution for the P&L in a week (5 days). The distribution of the P&L is given in Figure 7.1 .
The VaR at risk at the 95% confidence level is VaR95 = -71.41 and the expected shortfall is ES95 = -84.55. So we expect the book of options to lose at least 71.41 from its initial value 5% of the time.